Note that there appears to be an additional non-Newtonian term here. We need to calculate the inverse matrix to gu'v' of course.

OK I did the calculation and lo and behold g^1'0' = 0 exactly! There is NO "additional Non-Newtonian term" in this simple example. Such a term would mar the beauty of Einstein's general theory in this simple Galilean limit analogous to doing the simple hydrogen atom in the quantum theory of atoms.

Note that in this simple problem the inertial artificial gravity force felt by the non-inertial observer in globally flat spacetime is a pure "gravimagnetic" effect in terms of Einstein's local geometrodynamics in which there are no Newtonian gravity forces acting at a distance.